On Burnett coefficients in periodic media

Transcription

1 On Burnett coefficients in periodic media Carlos Conca a Departamento de Ingeniería Matemática, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, Santiago, Chile and Centro de Modelamiento Matemático, UMR 207, CNRS-UChile, Casilla 70/3 - Correo 3, Santiago, Chile Rafael Orive b Departamento de Matemáticas, Facultad de Ciencias, Universidad Autónoma de Madrid, Madrid, Spain Muthusamy Vanninathan c TIFR Center, P.O. Box 234, Bangalore-56002, India The aim of this work is to demonstrate a curious property of general periodic structures. It is well known that the corresponding homogenized matrix is positive definite. We calculate here the next order Burnett coefficients associated with such structures. We prove that these coefficients form a tensor which is negative semidefinite. We also provide some examples showing degeneracy in multidimension. I. INTRODUCTION The aim of this paper is to demonstrate a curious property of general periodic structures. Let us consider acoustic wave propagation in a periodic medium with a small period denoted by. To first order, we can approximate this medium by the associated homogenized medium. It is well known that the acoustic wave propagation in the homogenized medium provides good approximation to the propagation of sufficiently long waves in the original periodic medium Refs. and 9. Here, we are interested in higher order approximation. From our previous work, 2 it is known that the homogenized medium is described by the second order derivatives of the first Bloch eigenvalue at =0 and the next order approximation to the periodic medium is provided by a fourth order tensor, namely the fourth derivatives of at =0. In the physics literature such higher order derivatives are known as Burnett coefficients and they are of great interest Refs. 4 and 8. It is known that the homogenized matrix is positive definite. This work is devoted to the study of the next order approximation. The approximation of the periodic medium comes about from the asymptotic expansion of near =0, = 2! ! Substituting =, we get the asymptotic expansion of the first eigenvalue associated to the -periodic operator A near =0, a Electronic mail: cconca@dim.uchile.cl b Electronic mail: rafael.orive@uam.es c Electronic mail: vanni@mat.tifrbng.res.in

2 Conca, Orive and Vanninathan = 2 = 2! ! It is already established that the Hessian matrix 0/2 coincides with the homogenized matrix denoted by q k defined for instance in Ref. and hence is positive definite. In this write-up, we show that 4! " R N. Further, we show that there can be directions 0 such that 4! =0. However, in one dimension, as we show below, such a degeneracy cannot happen unless the medium is homogeneous. Before starting our computations, let us interpret the above result in terms of acoustic wave propagation in the original -periodic medium. From the above expansion, it is clear that 2! 0 2 if 2 4 is small. This shows that the usual homogenized medium, as remarked above, provides a good description provided the waves are long. However, for short waves the above approximation is poor. Indeed if 2 4 =O and 4 6 =o then we have 2! ! The above picture shows that long waves experience hyperbolic effects while short waves in question undergo some dispersion too. This dispersive nature medium is described by the fourth order tensor 4 0. However, the dispersion is not classical. It has a negative sign and we may then call it negative dispersion. Strictly speaking, the corresponding initial value problem modeling the propagation of such short waves is not well posed. We would like to bring to the attention of homogenization community that some curious materials with negative refraction, negative reflection coefficients are being conceived and produced see Refs. 5, 0,, and 3. Viewed in this light, our result says that a fine periodic structure provides one such curious material as far as short wave propagation is concerned. At this point, we would like to emphasize other features of our result. It came as a surprise to us to see a definite sign for the fourth order derivative, as it was not expected. Though our computations follow a general pattern, it is not clear to us whether higher order derivatives too have a definite sign. The next remark is concerned with the level of generality at which we are working: we have no restriction whatsoever on the original periodic medium except the classical ones. In other words, the material components used in mixing are arbitrary, their proportions are arbitrary and the microgeometry of mixing is also arbitrary. This contrasts sharply with the current efforts in producing curious or smart materials using particular components and following a particular design, for example, photonic crystals see Refs. 7, 6, and 2. II. PRELIMINARIES Let us now introduce the problem to be studied in this work. First, we remark that the summation with respect to the repeated indices is understood throughout this paper. We consider the operator

3 On Burnett coefficients def A = k y y R y ka y, N, 2. where the coefficients satisfy a k L # Y where Y = 0,2 N, i.e., each a k is a Y-periodic bounded measurable function defined on R N, and $ 0 such that a k y k 2 " R N, y Y a.e., a k = a k " k, =,...,N. For each 0, we consider also the -periodic operator A where 2.2 def A = x k ka x x with a k def x = a k x, x R N. 2.3 In homogenization theory, it is usual to refer to x and y as the slow and the fast variables, respectively. They are related by y=x/. Our results are based on Bloch waves associated with the operator A which we define now. Let us consider the following spectral problem parametrized by R N : find =R and =y; not zero such that A ; = ; in R N, ; is ;Y -periodic, i.e., y +2m; = e 2im y; " m Z N, y R N. 2.4 Next, by the Floquet theory, we define y;=e iy y; and 2.4 can be rewritten in terms of as follows: A = in R N, is Y-periodic. 2.5 Here, the operator A is called the translated operator and is defined by A = e iy Ae iy. It is well known see Refs. and 3 that for each Y= /2,/2 N, the above spectral problem 2.5 admits a discrete sequence of eigenvalues and their eigenfunctions referred to as Bloch waves introduced above enable us to describe the spectral resolution of A an unbounded self-adjoint operator in L 2 R N in the orthogonal basis e iy m y;m,y. To obtain the spectral resolution of A, let us introduce Bloch waves at the -scale, m = 2 m, m x; = m y;, m x; = m y;, where the variables x, and y, are related by y=x/ and =. Observe that m x; is Y-periodic in x and Y periodic with respect to. In the same manner, m ; is ;Y periodic because of the relation m x;=e ix m x;. Note that the dual cell at -scale is Y and hence we take to vary in Y in the sequel. With these notations, we have see Ref. the following. Theorem 2.: Let gl 2 R N. The mth Bloch coefficient of g at the -scale is defined as follows: B m g =R N gxe ix m x;dx " m, Y. Then the following inverse formula and Parseval s identity hold:

4 Conca, Orive and Vanninathan gx B = m ge ix m x;d, Y m= gx R 2 dx B = m g 2 d. N Y m= Finally, for all g in the domain of A, we get A gx = m B m ge ix m x;d. Y m= In the homogenization process, one can neglect all the modes for m2 see Refs. 4 and 2.To this end, we consider a sequence of solutions u of the equation A u = f in R N. 2.6 We can show that the component of u in the higher Bloch modes are not significant. More precisely, let us consider v defined by v x B = m u e ix m x;d. Y m=2 2.7 which is nothing but the projection of u corresponding to all higher Bloch modes. Then the following estimates on v are derived in Ref. 2. Proposition 2.2: Depending on the regularity of the source term f in 2.6, we have i If f H R N : v L 2 R N c f H R N. ii If f L 2 R N : v L 2 R N c 2 f L 2 R N. iii If f L 2 R N : v H R N c f L 2 R N. iv If f H R N : v L 2 R N c 3 f H R N. v If f H R N : v H R N c 2 f H R N. Here, we denote by H R N the seminorm of H R N. The above result is at the basis of neglecting higher order Bloch eigenvalues m m2 in the context of our discussion in the Introduction. III. FOURTH ORDER TENSOR 4 0 In this section, we present the expression for the fourth order tensor 4 0 and show that it is negative semidefinite. Recall that and ; depend analytically on in a small neighborhood B of =0 see Ref. 4. A. Derivatives of the first Bloch eigenvalue and eigenvector The purpose of this section is to present expressions for derivatives of the first Bloch eigenvalue and the first Bloch eigenvector ; at =0 and indicate a systematic method to compute them. For details of these computations, the reader is referred to Ref. 2. Our approach exploits the connection between the Bloch space computation with the multiscale computation. The derivatives of the first eigenvalue and eigenfunction in =0 exist thanks to the regularity property established in Ref. 4. In fact, we know that there exists 0 such that the first eigenvalue is an analytic function on B =, and there is a choice of the first eigenvector y; satisfying

5 On Burnett coefficients ; H # Y is analytic on B, y;0 = Y /2 = 2 N/2. Apart from the above result of regularity on the Bloch spectrum, the following proposition was also proved in Ref. 2. Proposition 3.: We have the relations 0 =0, D k 0 = k 0 =0 " k =,...,N, 2 D 2 k 0 = 2 0 = q k 2 k where q k are the homogenized coefficients defined by " k, =,...,N, q k = M Ya k + a km y m def = k + a km " k, =,...,N, 3. a y mdy with test function k defined by the following cell problem: A k = a k y in R N, Moreover, all odd order derivatives of at =0 vanish, i.e., k H # Y, M Y k = D 0 =0 " Z + N, odd. All even order derivatives of at =0 can be calculated in a systematic fashion. For instance, the fourth order derivatives have the following expressions: for all k,,m,n=,...,n, 4! D 4 kmn 0 = C n km + C k mn + C mnk + C m nk dy 4 Here, the operator C k is defined by a k q k mn + a m q m nk + a mn q mn k 3! + a nk q nk m + a km q km n + a n q n km dy. def C k = a kj y a kj y y j y j 3.3 is skew-adjoint, C k * = C k, and k, km are the test functions defined by the following cell problems: A k = a k q k 2 C k + C k in R N, k H # Y, M Y k =0, 3.4

6 Conca, Orive and Vanninathan A km = 3 a k q k m + a m q m k + a mk q mk 3 C k m + C mk + C m k in R N, 3.5 km H # Y, M Y km =0. The above expressions are obtained by differentiating the eigenvalue problem, A ; =0, and using that the branch ; can be so chosen that the following conditions are satisfied simultaneously: B ; H # Y is analytic, ; L 2 Y = " B, ImY y;dy =0 " B. B. D 4 0 is negative semidefinite First, we denote the fourth derivatives as B kmn = 4! D 4 kmn 0. Thus, by the Proposition 3., for any R N we get B kmn k n m = C n n km k m + C k k mn m n dy 4 + C mnk m n k + C m m nk k n dy 4 k a k q k mn m n + m a m q m nk k n dy 3! m n a mn q mn k k + k n a nk q nk m m dy 3! m k a km q km n n + m a n q n km m k dy. 3! Now, we introduce the following notations: C = n C n, = k k, 2 = k k, 3 = k m km, a = k m a km, q = k m q km. 3.6 Then, by the summation, the above expression is simplified to the following:

7 On Burnett coefficients B kmn k m n = C 3 dy a q 2 dy. 3.7 Since the test function k satisfies the cell problem 3.2, we have by the notations 3.6, A = k a k y. 3.8 Analogously, by the cell problems 3.4 and 3.5, we have the following Y-periodic problems: A 2 = a q C, 3.9 A 3 = a q C 2. By the notation 3.6, since C k is defined in 3.3, weget 3.0 Using 3.8 and integrating by parts, we have C 3 dy = 3 k a k dy. y C 3 dy = A 3 dy = A 3 dy = a q 2 C 2 dy, by 3.0. Therefore, from 3.7, we obtain B kmn k m n = a q 2 dy C 2 dy a q 2 dy. Again, by definition of C, we have C * = C and hence B kmn k m n = a q 2 dy a q C 2 dy. Using now 3.9, weget Substituting, B kmn k n m = a q 2 dy A 2 2 dy. we can rewrite the above expression as follows: B kmn k m n = a q = A 2 + C, A dy C 2 dy. A 2 2 dy 3. We show now that the last two terms of the above expression add up to zero. More precisely, we prove

8 Conca, Orive and Vanninathan A 2 2 dy + C 2 dy = To this end, let us first consider the last term. By definition of C, C Y 2 dy = 2 ky Via a simple integration by parts, we obtain a k y 3 3dy ky a k y 3 dy. C Y 2 dy = A 3 dy Y On the other hand, regarding the first term in 3.2, we can establish a more general relation, p + 2 A Y p+ p+ dy = A 2p+ dy " p N p +Y This proof is simply obtained by writing the expression in two different ways, namely a k y a k y y k 2p y k 2p+ 2p + and a k y p + p+ y k p + p+. A simple integration of these expressions leads us to the above relation 3.4. Finally, taking p = in 3.4 and using 3.3, weget3.2. Thus, we conclude the proof of the following result. Proposition 3.2: The tensor of fourth derivatives of in 0 is negative semidefinite. More precisely, for any R N, we get 2 4! D 4 kmn 0 k m n = A dy C. One-dimensional case In this case, we get a more simple formula for the form associated with the fourth order derivatives of. More exactly, we show that, for any R, we have 4! D4 0 4 = q dy Indeed, by 3.7 we know that 4! D4 0 4 = Then, we prove 3.6 by showing that 20 2 C 3 a q 2 dy C 3 dy q =0 2 + a q 2 dy. 3.7

9 On Burnett coefficients To prove the above result, we first establish some formulas where we use one-dimensional nature of the problem. Considering 3.8 and integrating, we get a d dy Now, considering 3.9 and integrating, we get = q a. 3.8 a d2 dy = a + c, where c is some constant. Dividing the above relation by a and integrating it over Y, wegetc = 0 and then the following relation results: d 2 dy =. 3.9 Because of the above relations 3.8 and 3.9, there are simplifications in one dimension which can be exploited to establish 3.7. To this end, it is enough to use the equations satisfied by and 3 and integration by parts. Remark 3.3: An immediate consequence of (3.6) is D 4 0 =0 Û ay is constant Indeed, if D 4 0=0 then =0 and by (3.8), the coefficient ay is constant. IV. DEGENERATE CASES Unfortunately, in several space dimensions 3.20 need not be true. It can happen that the coefficients a k are not constants and yet 4 D kmn 0 k m n = 0 for some In this section, we show some examples of such degenerates cases. First, we propose to prove two equivalent expressions for 4.. Proposition 4.: Let us pose where the constants 2 k are defined by 2 = 2 k k, 2 = k dy. 2! k 4.2 Then, if there exists R N 0 satisfying (4.), we get 2 = Also (4.), is equivalent to the following Hamilton-Jacobi type equation for : 4.3 a q +2 k a k y + a k y k y = Proof: From 3.5 and 4., we have immediately that =C, for some constant C. Integrating this relation and using the definition of and 2,weget

10 Conca, Orive and Vanninathan CY = 2 k Y k dy, and by definition 4.2 of 2 k, we obtain 4.3. Applying the operator A on the relation 4.3, weget A 2 = A 2 2. Since and 2 are solutions of 3.8 and 3.9, respectively, we have and A 2 = a q +2 k a k + A y A 2 2 = A a k y y k. We subtract the last two expressions and arrive at 4.4. Conversely, we can start from 4.4 and deduce 4.3. This completes the proof. Next, we present some examples of degenerate cases in several space dimensions. Case of laminates: We place ourselves in two dimensions. Consider the matrix of coefficients a k y, a y 0 0 a 22, with a 22 being a constant. Taking =0,, the following can be easily checked: 2 0, 0, q 22 = a 22, a = q = a 22. Recall that 2 is the solution of 3.2. Thus, the equation 4.4 and hence the property 4. are easily satisfied but a k y is not a constant matrix. ACKNOWLEDGMENTS This work has been partially supported by FONDAP through its Programme on Mathematical- Mechanics. The second author R.O. was partially supported by Grant No. S-0505/ESP/058 of the CAM Spain and Grants Nos. MTM and MTM of the MEC Spain. Bensoussan, A., Lions, J. L., and Papanicolaou, G., Asymptotic Analysis for Periodic Structures North-Holland, Amsterdam, Conca, C., Orive, R., and Vanninathan, M., Bloch approximation in homogenization and applications, SIAM J. Math. Anal. 33, Conca, C., Planchard, J., and Vanninathan, M., Fluids and Periodic Structures, RAM: Res. Appl. Math. 38 Wiley, Chichester, Masson, Paris, Conca, C., and Vanninathan, M., Homogenization of periodic structures via Bloch decomposition, SIAM J. Appl. Math. 57, Dong, Z. G., Zhu, S. N., Liu, H., Zhu, J., and Cao, W., Numerical simulations of negative-index refraction in wedgeshaped metamaterials, Phys. Rev. E 72, Felbacq, D., and Bouchitte, G., Homogenization near resonances and artificial magnetism from dielectrics, C. R. Acad. Sci., Ser. I: Math. 339, Felbacq, D., and Bouchitte, G., Negative refraction in periodic and random photonic crystals, New J. Phys. 7, Gaspard, P., Chaos, Scattering and Statistical Mechanics, Cambridge Nonlinear Science Series 9 Cambridge University Press, Cambridge, Santosa, F., and Symes, W. W., A dispersive effective medium for wave propagation in periodic composites, SIAM J. Appl. Math Shelby, R. A., Smith, D. R., and Schultz, S., Experimental verification of a negative index of refraction, Science 292,

11 On Burnett coefficients Smith, D. R., Pendry, J. B., and Wiltshire, M. C. K., Metamaterials and negative refractive index, Science 305, Wang, X., and Kempa, K., Negative refraction and subwavelength lensing in a polaritonic crystal, Phys. Rev. B 7, Wiltshire, M. C. K., Optical materials Bending light the wrong way, Science 292, van Beijeren, H., Transport properties of stochastic Lorentz models, Rev. Mod. Phys. 54,